Problem: When Harry divided $63$ by $h$, the remainder was $3$. When Harry divided $86$ by $h$, the remainder was $11$. What is the value of $h$?
Solution: The first statement is equivalent to the equation $63=mh+3$, and the second is equivalent to the equation $86=nh+11$, where $m$ and $n$ are integers. We simplify these to $mh=60$ and $nh=75$. Clearly, $h$ must be a factor of both $60$ and $75$. The possible values for $h$ are $1,3,5,15$ when considering only this condition, but in the problem, we cannot have a remainder of $11$ unless $h$ is greater than $11$. Thus, $h=\boxed{15}$.